186
DOC.
13
GENERALIZED THEORY OF RELATIVITY
gravitational equations.
Such
a
connection would have
to
exist insofar
as
the
gravitational equations
are
to
permit
arbitrary substitutions;
but in that
case,
it
seems
that
it
would be
impossible
to
find second-order differential
equations.
On the
other
hand,
if
it
were
established that the
gravitational equations permit only
a
particular
group
of
transformations,
then
it
would be understandable if
one
could
not
manage
with the differential
tensors
yielded by
the
general theory.
As has been
explained
in
the
physical part,
we are
not
able
to
take
a
stand
on
these questions.-
3.
On
the
Derivation
of the
Gravitational
Equations
The derivation of the
gravitational equations
described
by
Einstein
(Part
I, §5)
is
carried
out
step by step
in
the
following way:
We
start out
from the
term
that
is
definitely
to be
expected
in the
energy balance,
(47)
/
E
d
Jglaß
dy
MV
aßfiv
dxa
dxa
dxß
and reformulate
it
by integrating
by parts.16
In this
way
we
obtain
/
\
v=
E
a
\fgyaß
ÖYmv
d8fiv
-
E
/xv
d%v
aßfiv
^Xa
dxß
dxa
y
aßfiv
Uß
dxßxa
\
The first
sum on
the
right-hand
side has the desired form of
a
sum
of differential
quotients
and shall be denoted
by
A, so
that
we
have
(48)
\
^
=
E
a
'fsv
3v
/XV
aßfiv
dx
a
dxn
ß
dx
°
/
We
once
again integrate by parts
in
the second
sum on
the
right-hand
side. The
identity
will then take the form
U
-
A
E
a
yfg '
Y
5YMv 'dgfiv
+
E
dg,iV
a
\[g-
Y
ay
/XV
aßfiv a.Xa
aß
dxn
dx
aßfiv ÖXa
dx
aß
dx
\
ß
/
\
The first of the
sums
obtained
on
the
right-hand
side
can
be written
as a
sum
of
differentials and shall be denoted
by
(49)
*=
E
a
Jgyaß
aY^v dgfiv
I)y
dxß
dxa
aßfiv UAcr \
We differentiate
in
the second
sum.
Then
we
get
16
The derivation of the
identity we
are
seeking
becomes
simpler,
without
affecting
the
result,
if
we put
the factor
\fg
inside the differentiation
sign.